Question: If $f(x) = -\dfrac{1}{x},$ what is $f(f(f(f(f(6)))))$?
We see that $f(f(x)) = -\dfrac{1}{-\frac{1}{x}} = x$, therefore $f(f(f(f(f(6))))) = f(f(f(6))) = f(6) = \boxed{-\dfrac{1}{6}}.$